Problem Set - Differentiation

Expansions

For questions 1-9 suppose that $f x= c 0 + c 1 x-a + c 2 x-a 2 + c 3 x-a 3 + c 4 x-a 4 +…$

 Show that ${c}_{0}=f\left(a\right).$ Show that ${c}_{1}=\frac{df}{dx}{|}_{x=a}.$ Show that ${c}_{2}=\frac{1}{2}\left(\frac{{d}^{2}f}{d{x}^{2}}{|}_{x=a}\right).$ Show that ${c}_{3}=\frac{1}{3!}\left(\frac{{d}^{3}f}{d{x}^{3}}{|}_{x=a}\right).$ Show that ${c}_{4}=\frac{1}{4!}\left(\frac{{d}^{4}f}{d{x}^{4}}{|}_{x=a}\right).$ Show that ${c}_{5}=\frac{1}{5!}\left(\frac{{d}^{5}f}{d{x}^{5}}{|}_{x=a}\right).$ Explain why ${c}_{n}=\frac{1}{n!}\left(\frac{{d}^{n}f}{d{x}^{n}}{|}_{x=a}\right).$ Show that $f a+Δx =f a + df dx | x=a Δx + 1 2 d 2 f d x 2 | x=a Δx 2 + 1 3! d 3 f d x 3 | x=a Δx 3 + 1 4! d 4 f d x 4 | x=a Δx 4 +…$ Show that $\underset{\Delta x\to 0}{\mathrm{lim}}\frac{f\left(a+\Delta x\right)-f\left(a\right)}{\Delta x}=\frac{df}{dx}{|}_{x=a}.$ Give a series expansion for ${e}^{x}$ Give a series expansion for $\mathrm{sin}x$ Give a series expansion for $\mathrm{cos}x$ Give a series expansion for $\frac{1}{1-x}$ Give a series expansion for $\frac{1}{1+x}$ Give a series expansion for $\frac{1}{1+{x}^{2}}$ Explain why $1+\frac{1}{3}+\frac{1}{{3}^{2}}+\frac{1}{{3}^{3}}+\dots =\frac{3}{2}.$ Explain why $1+\frac{1}{3}+\frac{1}{{3}^{2}}+\frac{1}{{3}^{3}}+\dots +\frac{1}{{3}^{50}}=\frac{3}{2}-\frac{1}{2×{3}^{50}}.$

Derivatives at a point

 Let $y=\mathrm{tan}2x-2\mathrm{tan}x+2.$ Find $\frac{dy}{dx}$ at $x=\pi /4.$ Let $y=\frac{{\mathrm{sin}}^{2}x+\mathrm{cos}x}{1+{x}^{2}}.$ Find $\frac{dy}{dx}{|}_{x=0}$ and $\frac{dy}{dx}{|}_{x=\frac{\pi }{2}}.$ Let $y=\mathrm{cos}\left(\mathrm{sin}{x}^{2}\right).$ Find $\frac{dy}{dx}{|}_{x=\frac{\pi }{3}}.$ Let $y={\left(\mathrm{cot}\sqrt{x}+5{\mathrm{sin}}^{2}\sqrt{x}\right)}^{2}.$ Find $\frac{dy}{dx}$ at $x={\pi }^{2}/16.$ Let $y=\frac{\mathrm{sin}{x}^{2}}{\sqrt{1+{x}^{2}}}.$ Find $\frac{dy}{dx}{|}_{x=0}$ and $\frac{dy}{dx}{|}_{x=\sqrt{\frac{\pi }{2}}}.$

Differential equations

 If $y=\mathrm{tan}x+x$ show that ${\mathrm{cos}}^{2}x×\frac{{d}^{2}y}{d{x}^{2}}-2y+2x=0.$ If $y=A\mathrm{cos}nx+B\mathrm{sin}nx$ show that $>\frac{{d}^{2}y}{d{x}^{2}}+{n}^{2}y=0.$ If $y=2\mathrm{sin}x+3\mathrm{cos}x$ show that $y+\frac{{d}^{2}y}{d{x}^{2}}=0.$ If $y=a\mathrm{sin}x+b\mathrm{cos}x$ show that $y+\frac{{d}^{2}y}{d{x}^{2}}=0.$ If $y=\mathrm{sin}\left(\mathrm{sin}x\right)$ show that $y+\frac{{d}^{2}y}{d{x}^{2}}+\mathrm{tan}x\frac{dy}{dx}+y{\mathrm{cos}}^{2}x=0.$ If $y=a\mathrm{sin}x+b\mathrm{cos}x$ show that ${y}^{2}+{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{2}={a}^{2}+{b}^{2}.$ If $y=\sqrt{\mathrm{sin}x+\sqrt{\mathrm{sin}x+\sqrt{\mathrm{sin}x+\sqrt{\dots }}}}$ show that $\left(2y-1\right)\frac{dy}{dx}=\mathrm{cos}x.$

Parametric equations.

 Find $\frac{dy}{dx}$ when $x=a\mathrm{cos}\theta$ and $y=b\mathrm{sin}\theta .$ Find $\frac{dy}{dx}$ when $x=a\left(\theta +\mathrm{sin}\theta \right)$ and $y=a\left(1-\mathrm{cos}\theta \right).$ Find $\frac{dy}{dx}$ when $x=a{\mathrm{sec}}^{2}\theta$ and $y=b{\mathrm{tan}}^{3}\theta .$ Find $\frac{dy}{dx}$ when $x=b{\mathrm{sin}}^{3}\theta$ and $y=b{\mathrm{cos}}^{3}\theta .$ If $x=a\left(t-\mathrm{sin}t\right)$ and $y=a\left(1-\mathrm{cos}t\right)$ find $\frac{{d}^{2}y}{d{x}^{2}}.$ If $x=a\left(\theta +\mathrm{sin}\theta \right)$ and $y=a\left(1-\mathrm{cos}\theta \right)$ find $\frac{{d}^{2}y}{d{x}^{2}}$ at $\theta =\pi /2.$ If $x=2\mathrm{cos}\theta -\mathrm{cos}2\theta$ and $y=2\mathrm{sin}\theta -\mathrm{sin}2\theta$ find $\frac{{d}^{2}y}{d{x}^{2}}.$ If $x=\mathrm{sin}t$ and $y=\mathrm{sin}mt$ prove that $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{m}^{2}y=0.$

Implicit differentiation.

 Find $\frac{dy}{dx}$ when ${y}^{2}\mathrm{sin}x+y\mathrm{tan}x+\left(1+{x}^{2}\right)\mathrm{cos}x=0.$ Find $\frac{dy}{dx}$ when $\mathrm{sin}\left(xy\right)+x/y={x}^{2}-y.$ Find $\frac{dy}{dx}$ when $\frac{y}{1+{x}^{2}}+{\mathrm{tan}}^{2}x+\mathrm{sin}y=0.$ Find $\frac{dy}{dx}$ when $\mathrm{tan}\left(x+y\right)+\mathrm{tan}\left(x-y\right)=1.$ Find $\frac{dy}{dx}$ when $a\mathrm{sin}\left(xy\right)+b\mathrm{cos}\left(x,/,y\right)=0.$ If $x=\mathrm{ln}\left(\mathrm{tan}\left(y/x\right)\right)$ find $\frac{dy}{dx}.$

Derivatives with inverse trig functions.

 Find $\frac{dy}{dx}$ when $y={\mathrm{sin}}^{-1}{x}^{3}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\sqrt{x}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{sin}}^{-1}3x.$ Find $\frac{dy}{dx}$ when $y={\mathrm{csc}}^{-1}{x}^{2}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cos}}^{-1}\sqrt{x}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{csc}}^{-1}\left(\mathrm{sin}x\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\sqrt{x-1}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{sin}\left({\mathrm{tan}}^{-1}x\right).$ Find $\frac{dy}{dx}$ when $y=x{\mathrm{cos}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y=x{\mathrm{sin}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\sqrt{x}-{\mathrm{tan}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y=\left(1+{x}^{2}\right){\mathrm{tan}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y=\mathrm{tan}x{\mathrm{cos}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y=\frac{1}{2}\mathrm{ln}\left(\frac{1+x}{1-x}\right)+{\mathrm{tsn}}^{-1}x.$ Find $\frac{dy}{dx}$ when $\left(1-{x}^{2}\right){\mathrm{cos}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y=\mathrm{tan}x×{\mathrm{tan}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y={\mathrm{sec}}^{-1}x+{\mathrm{csc}}^{-1}x.$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\left(\frac{a}{x}\right)×{\mathrm{cot}}^{-1}\left(\frac{x}{a}\right).$ Find $\frac{dy}{dx}$ when $y={\left({\mathrm{tan}}^{-1}2x\right)}^{3}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cos}}^{-1}\left(\mathrm{tan}{x}^{2}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\left(\frac{2x}{1-{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{sec}}^{-1}\left(\frac{1-{x}^{2}}{1+{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\left(\frac{3x-{x}^{3}}{1-3{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{tan}}^{-1}\left(\frac{1+{x}^{2}}{1-{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{cos}}^{-1}\left(\frac{1-{x}^{2}}{1+{x}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y={\mathrm{cot}}^{-1}{\left(\frac{1+\mathrm{cos}x}{1-\mathrm{cos}x}\right)}^{\frac{1}{2}}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cot}}^{-1}{\left(\frac{1+\mathrm{cos}3x}{1-\mathrm{cos}3x}\right)}^{\frac{1}{2}}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{sin}}^{-1}{\left(\frac{a+b\mathrm{cos}x}{b+a\mathrm{cos}x}\right)}^{\frac{1}{2}}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cot}}^{-1}{\left(\frac{1+2\mathrm{cos}x}{2+\mathrm{cos}x}\right)}^{\frac{1}{2}}.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cot}}^{-1}{\left(\frac{1-\mathrm{cos}x}{\mathrm{sin}x}\right)}^{\frac{1}{2}}.$ Differentiate ${\mathrm{sin}}^{-1}\left(\frac{{x}^{2}-1}{1+{x}^{2}}\right)$ with respect to ${\mathrm{cos}}^{-1}\left(\frac{1-{x}^{2}}{1+{x}^{2}}\right)$ If $y=\frac{{\mathrm{sin}}^{-1}x}{\sqrt{1-{x}^{2}}}$ prove that $\left(1-{x}^{2}\right)\frac{dy}{dx}-xy=1.$

Derivatives with trigonometric functions

 Find $\frac{dy}{dx}$ when $y=x\mathrm{cos}x-\mathrm{sin}x.$ Find $\frac{dy}{dx}$ when $y={\mathrm{cos}}^{3}3x.$ Find $\frac{dy}{dx}$ when $y={\left({x}^{2}+\mathrm{cos}x\right)}^{4}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{sin}x\mathrm{sin}2x.$ Find $\frac{dy}{dx}$ when $y=\frac{\mathrm{sin}2x}{{x}^{2}}.$

Derivatives with exponentials and logs

 Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\left(x+\sqrt{{x}^{2}+{a}^{2}}\right).$ Find $\frac{dy}{dx}$ when $y=\frac{1+{e}^{x}}{1-{e}^{x}}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\left(\frac{{x}^{2}+x+1}{{x}^{2}-x-1}\right).$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\left[{e}^{x}{\left(\frac{x-2}{x+2}\right)}^{\frac{3}{4}}\right].$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\mathrm{ln}\mathrm{ln}{x}^{4}.$

Derivatives with exponentials, logs and trig functions.

 Find $\frac{dy}{dx}$ when $y={a}^{\mathrm{cos}1}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\frac{{\mathrm{sin}}^{m}x}{{\mathrm{cos}}^{n}x}.$ Find $\frac{dy}{dx}$ when $y={e}^{ax}\mathrm{sin}bx.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\left(\frac{1-\mathrm{cos}x}{1+\mathrm{cos}x}\right).$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\sqrt{\frac{1-\mathrm{tan}x}{1+\mathrm{tan}x}}.$ Find $\frac{dy}{dx}$ when $y={e}^{ax}\mathrm{cos}\left(bx+c\right).$ Find $\frac{dy}{dx}$ when $y=\frac{\sqrt{x+\mathrm{ln}\mathrm{tan}x}}{x{e}^{2x}}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\frac{1+x\mathrm{sin}x}{1-x\mathrm{sin}x}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}{\left(\frac{1-\mathrm{cos}x}{1+\mathrm{cos}x}\right)}^{\frac{1}{2}}.$ Find $\frac{dy}{dx}$ when $y=\mathrm{ln}\sqrt{\frac{1+\mathrm{sin}x}{1-\mathrm{sin}x}}.$ If $y=\mathrm{ln}\left(\mathrm{sin}x\right)$ show that $\frac{{d}^{3}y}{d{x}^{3}}=2{\mathrm{csc}}^{2}x\mathrm{cot}x.$ If $y={e}^{ax}\mathrm{cos}bx$ show that $\frac{{d}^{2}y}{d{x}^{2}}-2a\frac{dy}{dx}+\left({a}^{2}+{b}^{2}\right)y=0.$ If $y=a\mathrm{cos}\left(\mathrm{ln}x\right)+b\mathrm{sin}\left(\mathrm{ln}x\right)$ show that ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+y=0.$ If $y=A{e}^{-kt}\mathrm{cos}\left(pt+c\right)$ show that $\frac{{d}^{2}y}{d{t}^{2}}+2k\frac{dy}{dt}+{n}^{2}y=0,$ where ${n}^{2}={p}^{2}+{k}^{2}$ If $y={e}^{-x}\mathrm{cos}x$ show that $\frac{{d}^{4}y}{d{x}^{4}}+4y=0.$